Tropical Diffusion Dynamics: Why Nearly Identical Diffusion States Reach Different Output Routes

This paper investigates how deterministic probability-flow ordinary differential equations in diffusion models can produce different terminal outputs from nearly identical intermediate states. We address this by framing Gaussian denoising as a soft-minimum competition among clean-data routes, whose zero-noise limit yields nearest-support cells and, for finite atomic priors, tropical power cells. We prove that isolated regular two-route boundaries form a logistic transition layer of width $O({\sigma }^2)$ with peak denoiser sensitivity $O({\sigma }^{-2})$, where $\sigma$ is the noise standard deviation. We show that between-route posterior covariance simultaneously determines the Fisher tensor of route probabilities, the route component of reverse-flow strain, and the rate of Gaussian route-information gain. Because the remaining flow transports and amplifies these contrasts before decoding, we introduce a matrix-free Cauchy–Green Jacobian–vector product probe that maximizes linearized terminal displacement within a low-rank subspace. These theoretical insights are operationalized into Tropical Route Control (TRC), a modular framework combining finite-horizon probes, trajectory-level conformal event control, and exact prefix allocation under a hard continuation budget. Our experiments verify the predicted local sensitivity and finite-horizon perturbation effects and demonstrate that route-aware verification provides a valid and practical way to detect declared endpoint changes under a fixed computation budget in real settings.